Representation of a functor

A representation of $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ together with a universal element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, it is a specified natural isomorphism $\mathcal{C}(A,-)\cong F$.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.

If $H:{\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}\to \mathbf{Set}$ has chosen representations ${\psi }_B:\mathcal{A}(-,GB)\cong H(-,B)$, the Yoneda lemma determines a unique functor $G$ on morphisms by $\mathcal{A}(-,Gb)={\psi }_{B^{\prime }}^{-1}H(-,b){\psi }_B$. This makes the representing natural isomorphisms natural in $B$.

The universal element in a representation of a functor is the image of $1_A$ under the representing natural isomorphism. Any two universal elements induce unique mutually inverse maps between their representing objects, so representations are unique up to unique compatible isomorphism.